reserve x,y for object,X,Y for set,
  D for non empty set,
  i,j,k,l,m,n,m9,n9 for Nat,
  i0,j0,n0,m0 for non zero Nat,
  K for Field,
  a,b for Element of K,
  p for FinSequence of K,
  M for Matrix of n,K;

theorem Th8:
  for M be lower_triangular Matrix of n,K holds Det M=(the multF of
  K) $$ diagonal_of_Matrix M
proof
  let M be lower_triangular Matrix of n,K;
A1: Det M = Det M@ by MATRIXR2:43;
  M@ is upper_triangular Matrix of n,K by Th2;
  hence Det M= (the multF of K) $$ diagonal_of_Matrix M@ by A1,Th7
    .=(the multF of K) $$ diagonal_of_Matrix M by Th3;
end;
